Abstract
In this paper, we study the problem of estimating a multivariate normal covariance matrix with staircase pattern data. Two kinds of parameterizations in terms of the covariance matrix are used. One is Cholesky decomposition and another is Bartlett decomposition. Based on Cholesky decomposition of the covariance matrix, the closed form of the maximum likelihood estimator (MLE) of the covariance matrix is given. Using Bayesian method, we prove that the best equivariant estimator of the covariance matrix with respect to the special group related to Cholesky decomposition uniquely exists under the Stein loss. Consequently, the MLE of the covariance matrix is inadmissible under the Stein loss. Our method can also be applied to other invariant loss functions like the entropy loss and the symmetric loss. In addition, based on Bartlett decomposition of the covariance matrix, the Jeffreys prior and the reference prior of the covariance matrix with staircase pattern data are also obtained. Our reference prior is different from Berger and Yang’s reference prior. Interestingly, the Jeffreys prior with staircase pattern data is the same as that with complete data. The posterior properties are also investigated. Some simulation results are given for illustration.
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References
Anderson T.W. (1957). Maximum likelihood estimates for a multivariate normal distribution when some observations are missing. Journal of the American Statistical Association 52, 200–203
Anderson T.W. (1984). An introduction to multivariate statistical analysis. Wiley, New York
Anderson T.W., Olkin I. (1985). Maximum-likelihood estimation of the parameters of a multivariate normal distribution. Linear Algebra and its Applications 70, 147–171
Bartlett M.S. (1933). On the theory of statistical regression. Proceedings of the Royal Society of Edinburgh 53, 260–283
Berger J.O., Bernardo J.M. (1992). On the development of reference priors. Proceedings of the Fourth Valencia International Meeting, Bayesian Statistics 4, 35–49
Berger J.O., Strawderman W., Tang D. (2005). Posterior propriety and admissibility of hyperpriors in normal hierarchical models. The Annals of Statistics 33, 606–646
Brown P.J., Le N.D., Zidek J.V. (1994). Inference for a covariance matrix. In: Freeman P.R., Smith A.F.M. (eds). Aspects of uncertainty. A tribute to D. V. Lindley. Wiley, New York
Daniels M.J., Pourahmadi M. (2002). Bayesian analysis of covariance matrices and dynamic models for longitudinal data. Biometrika 89, 553–566
Dey D.K., Srinivasan C. (1985). Estimation of a covariance matrix under Stein’s loss. The Annals of Statistics 13, 1581–1591
Eaton, M. L. (1970). Some problems in covariance estimatio (preliminary report). Tech. Rep. 49, Department of Statistics, Stanford University.
Eaton M.L. (1989). Group invariance applications in statistics. Institute of Mathematical Statistics, Hayward
Gupta A.K., Nagar D.K. (2000). Matrix variate distributions. Chapman & Hall, New York
Haff L.R. (1991). The variational form of certain Bayes estimators. The Annals of Statistics 19, 1163–1190
Henderson H.V., Searle S.R. (1979). Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. The Canadian Journal of Statistics 7, 65–81
Jinadasa K.G., Tracy D.S. (1992). Maximum likelihood estimation for multivariate normal distribution with monotone sample. Communications in Statistics, Part A – Theory and Methods 21, 41–50
Kibria B.M.G., Sun L., Zidek J.V., Le N.D. (2002). Bayesian spatial prediction of random space-time fields with application to mapping PM2.5 exposure. Journal of the American Statistical Association 97, 112–124
Kiefer J. (1957). Invariance, minimax sequential estimation, and continuous time process. The Annals of Mathematical Statistics 28, 573–601
Konno Y. (2001). Inadmissibility of the maximum likelihood estimator of normal covariance matrices with the lattice conditional independence. Journal of Multivariate Analysis 79, 33–51
Little R.J.A., Rubin D.B. (1987). Statistical analysis with missing data. Wiley, New York
Liu C. (1993). Bartlett’s decomposition of the posterior distribution of the covariance for normal monotone ignorable missing data. Journal of Multivariate Analysis 46, 198–206
Liu C. (1999). Efficient ML estimation of the multivariate normal distribution from incomplete data. Journal of Multivariate Analysis 69, 206–217
Magnus J.R., Neudecker H. (1999). Matrix differential calculus with applications in statistics and econometrics. Wiley, New York
McCulloch C.E. (1982). Symmetric matrix derivatives with applications. Journal of the American Statistical Association 77, 679–682
Ni S., Sun D. (2003). Noninformative priors and frequentist risks of bayesian estimators of vector-autoregressive models. Journal of Econometrics 115, 159–197
Pourahmadi M. (1999). Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation. Biometrika 86, 677–690
Pourahmadi M. (2000). Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika 87, 425–435
Sun D., Sun X. (2005). Estimation of the multivariate normal precision and covariance matrices in a star-shape model. Annals of the Institute of Statistical Mathematics 57, 455–484
Yang R., Berger J.O. (1994). Estimation of a covariance matrix using the reference prior. The Annals of Statistics 22, 1195–1211
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Sun, X., Sun, D. Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data. AISM 59, 211–233 (2007). https://doi.org/10.1007/s10463-006-0044-x
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DOI: https://doi.org/10.1007/s10463-006-0044-x